Integrand size = 24, antiderivative size = 138 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]
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Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {3674891 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}}-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac {30371 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120}+\frac {334081 \sqrt {1-2 x} \sqrt {5 x+3}}{51200} \]
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {251}{100} \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx \\ & = -\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {2761}{320} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx \\ & = -\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {30371 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{1280} \\ & = -\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {334081 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{10240} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-3762261-4115415 x+16522420 x^2+37765600 x^3+33936000 x^4+11520000 x^5\right )-11024673 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1536000 \sqrt {3+5 x}} \]
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Time = 1.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\left (2304000 x^{4}+5404800 x^{3}+4310240 x^{2}+718340 x -1254087\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{153600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3674891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (46080000 x^{4} \sqrt {-10 x^{2}-x +3}+108096000 x^{3} \sqrt {-10 x^{2}-x +3}+86204800 x^{2} \sqrt {-10 x^{2}-x +3}+11024673 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+14366800 x \sqrt {-10 x^{2}-x +3}-25081740 \sqrt {-10 x^{2}-x +3}\right )}{3072000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
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Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{153600} \, {\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3674891}{1024000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {3}{2} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {539}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1121}{384} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30371}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3674891}{1024000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30371}{51200} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).
Time = 0.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{2560000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {351}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Timed out. \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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